advanced modeling of pellet- cladding interaction
TRANSCRIPT
DEGREE PROJECT IN ENGINEERING PHYSICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2016
Advanced Modeling of Pellet- Cladding Interaction
AGNIESZKA GOJAN
KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES
Advanced Modeling of Pellet-Cladding Interaction
AGNIESZKA GOJAN
Master of Science Thesis KTH Royal Institute of Technology
Stockholm, Sweden, 2016
Abstract A BWR fuel rod of Westinghouse design was used as a subject of cladding structural verification. Two computational tools were used to simulate a pellet-cladding interaction (PCI) phenomenon – ANSYS Workbench and a Westinghouse internal fuel rod code, STAV7. Additionally, advantages of using a cladding liner, made of Zr slightly alloyed with Sn, were presented. In order to illustrate a PCI event, a power ramp was simulated at burnup 35.6 MWd/kgU. Prior to that, an analysis for 0.0 MWd/kgU was performed in order to verify PCI behavior for fresh fuel (without swelling effects). The mechanical properties of the irradiated materials used and input data for the simulation tool are calculated theoretically on the basis of experimentally obtained data for unirradiated samples. The results show relatively similar results for ANSYS and for STAV7. The differences are caused by differences in the models, such as clad thermal expansion, pellet-clad bonding or pellet stiffness. ANSYS simulation for cladding without liner confirms the prediction of higher (<100 MPa) stresses on the clad inner surface for this configuration in comparison with liner cladding. Stress level on the outer surface is not affected. ANSYS simulation enables to capture the effect of pellet chamfer on the stress distribution in the cladding.
Keywords
AOO Anticipated Operational Occurrences
BWR Boiling Water Reactor
CFM Centreline Fuel Melt
FE Finite Element
LWR Light Water Reactor
MPS Missing Pellet Surfaces
NRC Nuclear Regulatory Commission
PCI Pellet Cladding Interaction
PCMI Pellet Cladding Mechanical Interaction
PMG Power Maneuvering Guidance
PWR Pressurized Water Reactor
SCC Stress Corrosion Cracking
List of appended papers
Appendix A
Fuel pellet material properties
Appendix B
Cladding material properties
CONTENTS
1 INTRODUCTION ________________________________________ 1
2 THEORETICAL BACKGROUND ______________________________ 2
2.1 PCI MECHANISM __________________________________________ 2 2.1.1 PCI/SCC failure ________________________________________ 2 2.1.2 Factors relevant for PCI __________________________________ 2
2.2 CLADDING LINER __________________________________________ 5 2.3 CRACK INITIATION AND PROPAGATION ___________________________ 5 2.4 POWER MANOUVERING _____________________________________ 7
3 METHODOLOGY ________________________________________ 8
3.1 MAIN PROJECT ASSUMPTIONS _________________________________ 8 3.2 STAV7 MODEL ____________________________________________ 9
3.2.1 Code description _______________________________________ 9 3.2.2 Power ramps _________________________________________ 10
3.3 ANSYS MODEL ____________________________________________ 11 3.3.1 Material properties ____________________________________ 12 3.3.2 Geometry and mesh ____________________________________ 13 3.3.3 Model assumptions _____________________________________ 15
3.4 COMPARISON OF STAV7 AND ANSYS MODELS _____________________ 18
4 RESULTS AND DISCUSSION _______________________________ 20
4.1 THERMAL SOLUTION ______________________________________ 20 4.2 STRUCTURAL SOLUTION ____________________________________ 20
4.2.1 ANSYS vs STAV7 ______________________________________ 21 4.2.2 Liner vs no liner _______________________________________ 24
5 CONCLUSIONS AND IDEAS FOR FUTURE RESEARCH ____________ 28
6 REFERENCES _________________________________________ 30
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1 INTRODUCTION
Release of highly radioactive fission products is undoubtedly the highest concern connected with nuclear power. For this reason, the integrity of fuel cladding must be preserved during the whole time of reactor operation. A dangerous threat for cladding failure in light water reactors (LWR) is the pellet-cladding interaction (PCI) phenomenon. The PCI failure can be a result of different mechanisms, such as stress corrosion cracking (SCC), hydrogen embrittlement ductile failure or delayed hydride cracking, or sometimes a combination of these mechanisms. Nevertheless, SCC is the most common failure mode since it is usually initiated the earliest, [1].
Although PCI can occur in all LWRs, it is especially common for boiling water reactors (BWR) as it is caused by abrupt power changes due to control rod movement. After a fuel rod failure, it is often difficult to be certain about the primary cause due to the rod degradation. It is usually safe to assume that if it occurs in fuel adjacent to a control rod pulled out of the core, the failure is a consequence of a PCI/SCC.
About a decade after the first PCI failures were reported in the early sixties, a thorough investigation began with the aim of understanding the phenomenon. As it was confirmed that PCI/SCC was responsible for a significant fraction of BWR fuel failures, there appeared an urgent need to find a remedy. The first approach considered power limitation during start-up and control rod manoeuvres in a way that would have ensured cladding stresses below a PCI threshold. Numerous power ramps were carried out in experimental reactors in order to establish the management recommendations for operators. This concept, although producing satisfying results in pressurized water reactors (PWR), caused significant capacity losses in BWRs. For this reason, a different solution was introduced, changing the approach onto design improvements. Two possibilities were considered, both in a form of material layers, bonded with the inner cladding surface. The first one was a thin (10 µm) iodine-resistant material, like copper, plated onto the Zircaloy-2 surface. The second – much thicker, a soft zirconium layer, metallurgically bonded to the cladding tube. It was found that, at higher burnups, copper forms an intermetallic layer with zirconium from the cladding that causes fuel degradation. All ramp tests with the Zr-liner had shown positive results and it was therefore introduced globally as a commercial product mitigating the PCI failure risk by the biggest nuclear companies. Since then, the classic PCI failures have been very infrequent, due to the combination of the beneficial effects of the power manoeuvring recommendations and the liner cladding.
Westinghouse has traditionally investigated the risk of PCI failure based on experimental data. While these experiments are extremely expensive, the idea of this work is to use high end software to investigate the PCI event. In this report, a comparison of software suitable for the analysis is conducted on the basis of different assumptions used. Detailed structural calculations of the response in cladding and liner are performed in order to help explaining the benefit of the liner. The future purpose is to make it even more effective by utilizing this knowledge.
In the first part of the work, a theoretical background on the PCI, liner and the related issues is introduced. The methodology section presents the software used. One evaluation
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is done in STAV7 – Westinghouse’s internal fuel performance code. The second tool is ANSYS Workbench – 3D finite element (FE) modeling software. Both calculation engines return a stress-strain distribution in a fuel rod with the liner cladding. The simulation without the liner is performed in ANSYS. The cladding is subjected to a power ramp typically known to cause risk of PCI failure. The choice of the input data is explained and the methodology of the analysis is described. In the last sections, the results of the simulation are presented and discussed, putting an emphasis on the differences in results and the role of the liner.
In an attempt to simulate a PCI event, appropriate assumptions and simplifications must be taken into consideration in order to represent a case as realistic as possible. It may transpire to be problematic due to the complexity of the process, which includes mechanical, chemical, thermodynamical and radiological interactions that must be analysed in different scales, [1].
2 THEORETICAL BACKGROUND
2.1 PCI MECHANISM
2.1.1 PCI/SCC failure
The PCI phenomenon can be defined as an increase in cladding stress due to expansion of the pellet against the cladding. It is usually a consequence of a power ramp, e.g. during a startup or control rods maneuvering, followed by a low power stage. Tensile stress/strain caused by the power increase in combination with chemically aggressive environment (especially iodine but also cesium and cadmium) and sensitive cladding material leads to SCC. This low strain failure mechanism attacks the cladding inner surface. Pellet-cladding mechanical interaction (PCMI) occurs when the cladding stresses due to pellet expansion become so large that it will break strictly due to mechanical action without the influence of the chemical agent (SCC is not the driving force).
Although no specified criterion exists for preventing cladding failures caused by PCI/SCC, U.S. NRC have proposed two related criteria in order limit its occurrences. They should be implemented for fuel rods during normal operation and condition II transients - Anticipated Operational Occurrences (AOOs):
• Centreline fuel melt (CFM) should not occur; • Ductility induced by a transient behaviour in highly irradiated zirconium alloys
should not exceed 1%.
PCI/SCC does not require any (or requires very little) plastic strain to occur. While the strain criterion increases the margin to a PCMI event, it does not ensure a PCI failure free operation since it can occur at much lower strains, [2].
2.1.2 Factors relevant for PCI
PCI phenomenon is a function of factors related to, [3]:
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• Fuel rod design and fabrication quality Pellet defects, for example missing pellet surfaces (MPS), are responsible for high local clad stress concentrations, thus their size and frequency of occurrence have an impact on PCI. The choice of fabricated pellet density (influence on swelling accommodation and densification) and of the initial distance between the pellet and the cladding (the gap width) has an effect on how fast (at what burnup level) the gap closes.
• Material susceptibility The process of material selection takes into account the future change of its characteristics due to exposure to irradiation and high temperatures. Choice of a material on the cladding inner surface (cladding liner) with different mechanical properties than the base material can reduce shear stresses during PCI. Fuel pellet cracking, densification and swelling are influenced by the choice of the fuel material.
• Operating conditions Both the local conditions (that are the consequence of core design) and the rate of power change influence the PCI margin. Higher nodal peak or conditioning power levels as well as local power change increase the pellet expansion which in turn increases the cladding stresses. Nodal burnup level affects pellet swelling and hence the moment of gap closure. Power ramp rate has an impact on the cladding stress relaxation and results in different cladding stresses at the end of the maneuver.
Some of the relevant mechanisms are described below in more detail.
2.1.2.1 Fuel densification, swelling and cracking
Fuel densification occurs due to rise in power and irradiation and it is a continuation of the sintering process, implemented during fabrication. The decrease in pellet volume is connected with the loss of the as-fabricated pores. If the initial density is increased, the densification process will be reduced which in turn may lead to an earlier onset on the gap closure. The swelling phenomena counter balances the densification with the remaining pores accommodating the gaseous fission products to some extent. Again, with the reduced porosity, the effects of swelling will lead to an earlier PCI.
Low thermal conductivity of UO2 has an impact on the temperature gradient through the pellet radius. Higher thermal expansion of the central region than in the periphery can cause radial cracks of the pellet, see Figure 1. The cracks divide the pellet into parts which can radially relocate into the pellet-clad gap. This results in local stress concentrations on the clad inner surface, [3].
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Figure 1: Schematic representation of the radial pellet cracking phenomena with a closed pellet-clad gap. Parts created due to cracks can relocate outwards and cause local stress peaks. Additionally, the cracks facilitate release of fission products and may influence crack nucleation on the cladding inner surface.
2.1.2.2 Fuel and clad creep
Both pellet and cladding will experience creep, a progressing in time change of shape as a response for an applied load. As the pellet expands, due to thermal expansion and swelling, it meets a radial constraint – the cladding. This interaction applies loads in both clad and pellet and causes stresses. Within a longer period, this stress induces cladding creep outwards and the pellet inwards. Two main effects that contribute to the total creep rate are the temperature and its irradiation state (as neutron flux enforces defects mobility in lattices), [3]. Since creep induces strain after longer times, it is able to release larger stresses between the pellet and the cladding. Also, if they are kept below a certain threshold, it can reduce the potential for SCC, [1]. The time scale of the creep relaxation during PCI can be in order of both months and hours, depending on the cladding hoop stress level. During a transient, cladding relaxation due to creep will occur much slower than the pellet thermal expansion that results in clad stress increase to relatively high levels. At high hoop stress, the creep relaxation will achieve a higher relaxation rate and after sufficiently long time - reach equilibrium, [1]. Nevertheless, for fast transients creep relaxation will have minor implications (when analysed in the time scale of the transient).
2.1.2.3 Pellet-clad gap parameters and interface bonding
The pellet-clad gap characteristics are the key concerns in the PCI study. The essential parameters that change during the power ramp duration are the gap thickness, gap gas heat conductance and pressurization. A transient fission gas release from the pellet changes the gap gas composition and hence its heat conductance which in turn affects the fuel temperature, [4]. The gap thickness decreases as the pellet expands and the gas accumulation in the reduced gap causes higher gas pressures. Even at the moment of the total gap closure, the gas remains present in the surface pores of the materials and in the clearances due to pellet chamfering.
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Due to fission fragments recoiled from the fuel, which include chemically aggressive elements, the chemistry and structure of the pellet-clad interface is changed. A possibility of formation of an interaction layer has been observed at higher burnup as the fuel bonds closely with the cladding inner surface. The bonding limits the relative movement of the fuel and clad in the circumferential direction and can lead to an increase of local stresses, [3]. With the addition of aggressive fission products contained in the layer, the margins to SCC are decreased.
2.2 CLADDING LINER
The liner serves as both mechanical and chemical barrier. As previously explained, the yield strength of a material plays an important role when it comes to crack initiation and propagation. Therefore, an additional material that covers the inner cladding surface must possess lower yield strength than Zircaloy-2 and hence be softer. The second purpose of the liner is to separate the susceptible Zircaloy-2 from the chemically aggressive iodine and cesium and decrease the probability of SCC.
A zirconium layer, metallurgically bonded to the clad base material, is softer and more ductile than Zircaloy-2 and hence fulfils the above mentioned functions. Nevertheless, pure zirconium is susceptible to corrosion and can cause axial splits of the inner cladding surface, therefore it is useless in PCI prevention. For this reason, the liner must contain a small quantity of another element, for example tin, [4]. The Zr-Sn liner of Westinghouse design has a thickness of ~70 µm.
2.3 CRACK INITIATION AND PROPAGATION
A possible cladding fracture will occur in the following steps: 1) crack initiation in the place of stress concentration, 2) crack propagation which rate and behaviour depends on the material ductility, 3) final failure, [6]. The failure can be avoided by preclusion of crack nucleation or by hindering the crack propagation. Cladding crack initiates at the inner surface, very often since the initial material flaws and progresses towards the outer in a minute scale, [1]. These pre-existing micro-defects are however not a necessary cause and their mere existence is not a sufficient factor for crack initiation, [7].
A combination of ductility decrease with an increase of yield strength makes the material more sensitive for cracking. Both yield strength and hardness indicate resistance of a material to plastic deformation and the relation between them can be approximated to proportional (for some metals more than the others), [6]. During irradiation, the inner surface of the cladding hardens significantly up to a certain thickness. Studies of the concentration of elements in this region and the amount of fission fragments recoiled from the fuel confirm that they are responsible for the change in the mechanical property of the material, [8].
The previously mentioned aggressive environment is a consequence of the release of corrosive fission products, especially iodine but also cesium and cadmium. They escape from the pellet through thermal cracking (radial cracks) or the diffusion process and
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interact with the cladding inner surface. This attack can be presented through the example of iodine reacting with zirconium forming iodides, [9]:
𝑍𝑟 + 2𝐼& ↔ 𝑍𝑟𝐼& + 𝐼& ↔ 𝑍𝑟𝐼( +12𝐼& ↔ 𝑍𝑟𝐼*
from which 𝑍𝑟𝐼* has the highest capability of causing SCC, [1]. Cesium by itself or together with iodine (CsI) does not attack Zircaloy. It does, however, combine with oxygen from the fuel and transfers the oxygen which dissolves in Zircaloy. Cadmium has a tendency to diffuse into the cladding material and form compounds at grain boundaries, [10].
When an irradiated cladding with a liner is subjected to PCI, the stress caused by the swollen pellet causes a micro crack in the oxide layer of the liner and uncovers bare liner material. Fission products accumulate on the fresh material which initiates a crack and it propagates as the fission products attack newly exposed bare liner, [2]. Each PCI event will uncover new fresh material surface which results in a cyclic response of the material.
According to tests [3], the crack reaches the same depth as the hardened part of the liner and it is illustrated in Figure 2. It means that the part of the material that remains soft still has the ability of sustaining the function of the liner and it is one of the reasons for the choice of its thickness. For a cladding not protected with liner, the crack propagation is not hindered and the crack can advance, see Figure 1. This is due to the different crack propagation mechanisms in relatively ductile and brittle materials. The mode of crack propagation in softer, more ductile materials will be described as stable – the crack length will increase relatively slowly with extensive plastic deformation in the crack neighboring area. The propagation will not continue without increase in the applied stress. The crack will not proceed through the remaining soft material because the stress at the tip of the crack is not sufficient for propagation in ductile material, it will however be oxidized when in contact with UO2, [4]. This is not the case with less ductile materials. The crack propagation mode in brittle materials is considered unstable due to its rapid and spontaneous nature – the crack will grow even when the value of applied stress will not increase any further, [6].
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Figure 2: Schematic representation of a crack propagation in the hardened part of the cladding liner. The liner layer hardens up to a certain thickness due to fission products recoiled from the fuel. The crack propagation is hindered by the part that remained soft.
A crack at the cladding inner surface will grow in the direction perpendicular to the applied tensile stress. In case of a majority of brittle materials, crack propagation (and possible eventual fracture) will be a consequence of breaking atomics bonds along a certain propagation plane which usually passes through the interior of material grains (transgranular crack), [6].
2.4 POWER MANOUVERING
In order to keep the safety margins in commercial reactors, power ramp tests are performed in research reactors in order to investigate the influence of different parameters, such as cladding resistance, burnup, conditioning and peak power, on the PCI failure probability. Ramp tests in different conditions together with successful power maneuvers history from commercial reactors, allow evaluation of power range of operation with low possibility of a PCI failure, [4].
The fundamental aspect of the power conditioning is to allow relaxation in the cladding before next step of power increase. The relaxation is caused by creep in the cladding that lowers the stress in the cladding and thereby makes it more resistant to next power step. Once conditioned to a certain level, power can be changed freely up to this value as long as the deconditioning was not allowed. Deconditioning takes place when a reactor operates under reduced power for a longer period which is usually a consequence of testing, maintenance or coastdown at the end of cycle, [3]. It decreases the cladding-pellet contact pressure which, in case of further power reduction, can result in re-opening of the hot gap. Deconditioning is a process much slower than conditioning (usually takes months while conditioning is measured in hours). The deconditioning rate depends on the level of power reduction and the time of operation at reduced power.
Power Maneuvering Guidance (PMG) is a list of recommendations provided by Westinghouse for the use of its customers in order to reduce the probability for a PCI
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failure. The PMG includes a maximum Linear Heat Generation Rate level, defined with consideration of the burnup history and the clad design and with a possibility of increasing this level by power conditioning. Furthermore, the power ramp rate above a certain power threshold must be performed either in a continuous way or in equivalent steps. PMG specifies also that deconditioning rate, as a function of burnup, must be evaluated with regard to creep and pellet swelling, [4].
3 METHODOLOGY
The main goal of the investigation is to demonstrate advantages of a cladding liner through simulations of power ramps that could have occurred in existing reactors. The idea is to compare the level of stresses and strains in the cladding for different configurations. The analysis is completed with the use of two calculation tools – ANSYS Workbench and STAV7. This approach enables conducting software comparison through the interpretation of the differences in results as well as by the assessment of their weak points.
The analysis is divided into two parts. The aim of the first case is to present and discuss the results from the ANSYS simulation for the cladding with and without a liner layer. In the second part, outcomes from the ANSYS model with liner cladding are compared with the STAV7 results. As the goal is to justify the discrepancies by differences in the structure of the two models, the relevant setting options must be determined. The Methodology section of the report focuses on the process of selection of the adequate study cases and describes both calculation engines, as well as the assumptions made.
3.1 MAIN PROJECT ASSUMPTIONS
In this study only a “classical” PCI failure is investigated – the fuel rod fabrication quality factor is not considered. The modelled version of the pellet does not take into account defects that can enhance the PCI mode, e.g. MPS defects. Another pellet defect possible to occur - radial cracking mechanism - can be a consequence of relatively low thermal conductivity of uranium dioxide and by consequence high thermal gradients across the pellet radius. Although this effect can have a big influence on pellet-clad gap/contact due to possible pellet fragment’s radial relocation, it is disregarded in order to remain simplicity of the model.
In order to present a severe case of a PCI, a fast power increase with short hold time is simulated. As the time does not play important role in the simulation, the time related material response, such as creep, is neglected. Swelling processes that occur during the short time of the simulation are not covered either, although the state of the pellet that results from swelling at investigated burnup levels are not disrespected.
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3.2 STAV7 MODEL
3.2.1 Code description
STAV7 is a code that predicts LWR fuel rod steady-state performance with the ability of simulating important phenomena that occur during irradiation. Its main purpose is verification of safety issues of fuel rod behaviour and is also used as a tool of post-radiation examination, [5]. For practical purposes, only fragments of the code that are significant for this project will be described.
The subject of analysis is the active height of a fuel rod divided into axial segments of equal length. Fuel pellets are considered as right circular cylinders – chamfer is not taken into consideration in the geometry design. Nonetheless, chamfering, dishing (see Figure 3) and stacking faults are accounted for the void volume definition.
Figure 3: A schematic representation of chamfering and dishing in the pellet design. These effects are not considered in the STAV7 geometry model.
STAV7 input includes geometry definition, as well as power history, selection of models and thermal-hydraulics data. Pellet and clad radii values are defined according to the drawings and number of rod axial nodes is chosen. Fuel rod power history is specified as average linear heat generation rate as a function of either burnup or time. Moreover, axial peaking factors are introduced. Coolant temperature is considered to be the saturated value for the stated pressure at all axial positions. The heat transfer between the cladding and the coolant is described by the correlation for nucleate boiling. The calculation includes the temperature step jumps over crud and cladding oxide and the cladding wall thinning due to oxide layer growth. A simplified version of STAV7 computation steps is presented in Figure 4.
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Figure 4: Simplified flow chart of STAV7 computational path.
Important quantities from PCI perspective are pellet-cladding gap or contact pressure. Fuel pellet densification, swelling and creep are taken into consideration and the contact between pellet and cladding is frictional. STAV7 output provides solutions after each time step for separate nodes. Relevant output parameters for this evaluation are radial temperature distribution in pellet and clad as well as stresses and strains, evaluated due to elastic, plastic, thermal and creep deformations.
3.2.2 Power ramps
A power ramp suitable for the simulation was chosen from a database of a Westinghouse’s licensing project and modified in order to meet the goals of this work. Two cases were selected: one at zero burnup with the aim to simulate the rod state without the effect of swelling and one in the fuel “middle life” – at an average burnup of 35.6 MWd/kgU. Ramp details are presented in Table 1. All input parameters are specified for standard BWR model with uranium dioxide fuel and Zircaloy-2 cladding, part length rods and rods with burnable absorbers are not used. “Best estimation” methodology is implemented in order to obtain less and more realistic results. The ramp evolution is investigated only up to a power level that represents the centerline temperature below fuel melt (with an adequate safety margin). Figure 5 presents axial power peaking factors for each of the 25 nodes (where node 1 is at the bottom of the rod). Maximum power level is achieved at nodes 9 and 17. Only node 17 will be subjected to the analysis and all nodal quantities in the results will be presented for this node.
Coolant thermal-‐hydraulics calculations
Cladding and pellet thermal analysis
Fission gas release calculations
Cladding mechanical response
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Figure 5: Example of axial peaking factors for all 25 axial nodes of the fuel rod used in the analysis.
Table 1: Characteristic values of the simulated power ramps.
Rod average burnup 0.0 MWd/kgU 35.6 MWd/kgU
Average terminal power level 53.4 kW/m 53.4 kW/m
Nodal terminal power level (node 17) 60 kW/m 60 kW/m
Nodal terminal pellet centerline temperature (node 17) 1995 ˚C 2413 ˚C
Fuel melting point 2805 ˚C 2691˚C
3.3 ANSYS MODEL
ANSYS Workbench is a 3D FEM computation tool suited for mechanical analysis of a fuel rod state. In order to create an ANSYS model that can provide a sufficiently realistic
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Axial power peaking factor
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representation of a PCI event in a reactor, it is crucial to decide which parameters are influencing the interaction. The ANSYS model requires definition of material data of each component, geometry and mesh specification as well as design of thermal and static structural models. The steady state thermal model computes the effects of the steady state thermal loads on the design by providing e.g. temperature distribution over the given geometry, [11]. The objective of the analysis is to simulate a power transient hence the ramp must be divided in steps to evaluate a steady-state solution after each of them. The static structural analysis calculates mechanical parameters, such as elastic and plastic strains, stresses or contact pressures, under static load conditions. The solutions can be coupled – outcome from the thermal model can be used as an input to the structural one.
3.3.1 Material properties
Fuel pellets, made of uranium dioxide, are considered in the ANSYS model as nearly stiff (although not perfectly stiff) bodies in direct contact with elastic and plastic cladding. For the sake of comparison of the Ansys model with STAV7, all material properties are evaluated in accordance with the STAV7 Model Description. Change of material characteristics due to irradiation is taken into account by introduction of a certain neutron fluence. A typical behavior of irradiated materials is presented in Figure 6 - irradiation leads to changes in the yield strength (by Δσ sector presented in Figure 6) and in the ultimate tensile strength, as well as reducing ductility. In other words, radiation hardening means strengthening of the material while causing it partial loss of ductility. It is a result of displacing atoms from their lattice sites initiated by a primary knock-on atom that causes a cascade of these effects in the irradiated material.
Since irradiation itself causes many displacements, the plastic region hardening, just after yielding, occurs at a lower rate than in case of an unirradiated material. In a fresh material, the defect density in the yielding regions is relatively low and the hardening process is fast until the point where there are too many dislocations already; it happens at relatively low strain levels (~𝜀∗ in Figure 6). Then the hardening process slows down due to high defect density. It is not the case with materials exposed to neutron fluence – many changes in the material are caused by neutrons themselves – the defect density is high and the hardening process is slow already from the yielding point. The fragments of plots where slow hardening occur, after 𝜀∗, are very similar for both unirradiated and irradiated materials – it can be assumed that the plots have the same slopes. This assumption will simplify the irradiated liner strain-stress evaluation.
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Figure 6: Strain-stress relation for unirradiated and irradiated material. After the strain level of ε* tangents of the two slopes can be approximated as equal.
Apart from solid fission products and fission gas bubble swelling, the main reason for the load induced on the cladding is pellet thermal expansion, [6]. The expansion process strictly follows pellet temperature and hence it will gradually increase even at constant power, if the power level has been raised. A temperature solution to be used as input to the structural analysis is described in detail in Appendix A.
The experimentally collected properties of zirconium-alloy – the base cladding material – mainly exist for unirradiated samples. In order to get more accurate results in the study, the change of properties due to irradiation is calculated theoretically and presented in Appendix B.
The liner is made of a relatively soft and ductile material, such as nearly pure zirconium, slightly alloyed, for example with tin. Like in case of the cladding, there is no irradiated strain-stress data for the liner available. However, the hardness of the material was measured in both normal and irradiated conditions and this information has been used to estimate the liner material properties at irradiated conditions. Bilinear Isotropic Hardening option is chosen as plasticity model for the liner for practical reasons – it requires much less calculation time than Multilinear option. It is considered as an acceptable estimation due to nearly linear development of stress with strain after yielding for irradiated materials (as seen in Figure 6). For this reason, the only input values needed are the yield strength and the tangent of the slope in different temperatures. The isotropic assumption is sufficient since only loading is simulated (no cyclic load, kinematic assumption is not needed). The comparison of yield stress changes with temperature for the base material and for the liner is presented in Appendix B.
3.3.2 Geometry and mesh
The initial geometry design consists of one and a half pellet in contact with each other, enclosed in a cladding, as shown in Figure 7. In order to approximate the state of the pellet at certain previously discussed burnups, pellet swelling must have been represented by increase of the pellet radius. By consequence, the pellet-clad gap is considered closed in the
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design. Moreover, chamfering is included in the pellet shape. Geometry is created in two options: with cladding divided in base material and liner, as well as an option with no separation. All component dimensions are set according to Westinghouse’s BWR fuel rod design.
Figure 7: Original geometry design in ANSYS (with a part of cladding removed for transparency reasons) - an axial fragment of a fuel rod.
In order to avoid too long calculation time, the subject of the analysis is considered to be just a slice of the original fuel rod design - a sector with an arc length of 10˚, presented in Figure 8. The use of a small slice is a common way of handling the axisymmetric assumption in a 3D model.
Figure 8: Geometry subjected to the analysis – a slice with a radial thickness of 10˚that includes one pellet and a half with the surrounding clad.
Mapped meshing option is used with quadrilateral recommendation and mesh densification in the vicinity of the pellets chamfer area, as presented in Figure 9.
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Figure 9: Mesh of the geometry subjected to analysis. Denser mesh is used in the cladding and pellet areas close to the pellet chamfer for better resolution.
3.3.3 Model assumptions
The model consists of a steady-state thermal and a static structural solutions coupled with each other in such way that the structural model is allowed to use the thermal results but not the other way around. Bonded connection is determined at the contact area between the two pellets. With this approximation there is no need for any further specification of the heat transfer between them. The contact between the pellets and cladding is set as frictionless. This assumption, used for practical reasons (model simplicity), is accurate enough to investigate strains and stresses in the circumferential direction.
3.3.3.1 Thermal model
There are two possible ways of handling the steady-state thermal model. The first possibility is entirely independent of STAV7 – pellet internal heat generation rate is defined for each specified step of the solution and all relevant temperatures of the pellet and the cladding are calculated by ANSYS. The second option consists of using the temperature solution from STAV7 as an input for the thermal analysis.
In the case of the first solution the issue of specifying heat transfer coefficients can become problematic. Both the film convection coefficient at the external clad surface and the thermal conductance coefficient at the cladding-pellet interface are evaluated through trial and error in order to achieve relevant corresponding temperatures similar to those from the literature of from STAV7 output (at the same power level). The film convection coefficient is approximated to 0.09 W/(mm2K) by comparison of internal and external cladding temperatures. The thermal conductance coefficient, estimated as 1.65 · 10-3
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W/(mm2K), is chosen in a way to match the value of pellet centerline temperature with the chosen reference. This method of approaching the thermal solution also requires calculating the outer clad temperature change due to crud and oxide layer formation. This can be done using relations from the STAV7 Model Description. The best way to employ it is to increase the defined coolant temperature by the obtained value of temperature difference.
The power ramp, designed for STAV7 and described in the section 3.2.2, is simplified as shown in Figure 10. The power levels 4 kW/m and 60 kW/m are based on the ramp in STAV7; the beginning point of the ramp at 0 kW/m is added for the ANSYS simulations purposes. Time is used exclusively to define the transient event; apart from that it has no meaning in the analysis. Several steps are defined to guarantee a complete solution during the whole ramp duration.
Figure 10: Simplification of the power ramp used as ANSYS input. The ramp includes solution steps in order to evaluate results after each of them.
The second option is possible to implement due to high accuracy of STAV7 temperature prediction. In order to achieve a similar (parabolic) temperature distribution through the pellet to the one from STAV7, the pellet radius is divided in uni-length sectors with the temperature specified at the end of each division. This approach guarantees no discrepancies in temperatures (e.g. due to badly chosen heat transfer coefficients) and allows to focus on the mechanical results, which is the main goal of the comparison. Therefore, this method is proceeded with in the project.
3.3.3.2 Structural model
The results from the simulation performed in the steady-state thermal model are used as input to the static structural solution. The next step is to define boundary conditions applied to the geometry. The pressure at the cladding external surface is set to 7.31 MPa, a typical value for a BWR, Figure 11. The external temperature, representing coolant, is defined as 289 ˚C (water boiling point at 7.31 MPa). Symmetry boundary conditions are specified for all the clad and pellet surfaces that result from slicing the pellet, see Figure 11. Frictionless support represents the previously mentioned axisymmetric assumption which guarantees everything that happens in the tangential direction being considered as
1 2
3
4
5
6
0 kW/m
ramp terminal level:60 kW/m
4 kW/m
17
constant (at any point on the perimeter of the initial geometry, Figure 7). Additionally, the clad and pellet surfaces marked with the letter E in Figure 11 will not deform in the axial direction (Z axis), although they can move in this direction as a whole.
Figure 11: Pressure on the clad external surface and symmetry boundary conditions in the ANSYS model.
As the created FE model does not capture the effects of solid and gaseous fission gas release from the fuel, the pellet-gap parameters will be different in Ansys and in STAV7. To compensate for this issue, the gap gas pressure values will be extracted from STAV7 for given burnup and used in ANSYS. The fission products are partially accommodated in pores in the fuel and cause pellet swelling which decreases the pellet-clad gap. Fission products in gaseous state are released to the gap and increase the pressure in the gap. This effect is captured in Figure 12 that shows that the rod internal gas pressure is higher for greater burnup value. The initial pressure value, i.e. the fill pressure, is usually set as 0.8 MPa. Another reason for the rise in pressure is increasing temperature during a power ramp. This is the main reason for the elevated value of the fresh fuel pressure in comparison with the as-fabricated value. Using the same pressure values in both simulations guarantees that the gap will be closed at the same moment.
Figure 12: Rod internal gas pressure as function of the nodal power for the two studied burnups – output from STAV7.
18
Another issue that is not captured by ANSYS is pellet swelling, densification and fuel parts radial relocation. All the effects influence the pellet radius and therefore the pellet-clad gap. Even though the pellet thermal expansion is modeled separately in ANSYS (see Appendix A), its effect at the ramp initial point at a certain burnup must be included in the initial total radius change. The results from STAV7 for 35.6 MWd/kgU are presented in Figure 13. No gaseous swelling is captured in the solution. Swelling and densification do not depend on the power ramp – they are constant at certain burnups. In comparison, for the fresh fuel both of the values are zero and the change of radius due to relocation is at a certain constant level during the whole ramp.
Figure 13: Pellet radius change caused by phenomena that result from irradiation and temperature effects as STAV7 outcome for 35.6 MWd/kgU burnup. Total radius change is a sum of the effects and its initial value was used as an input to ANSYS.
To compensate for the state of the pellet radius (and hence the gap width) caused by thermal expansion, swelling, densification and relocation, ANSYS can offset the contact surfaces by a specified value. This function is used in case of gaps between two bodies introduced in the initial geometry design in situations where the bodies can come in and out of contact with each other. It is the case for the frictionless behavior of the pellet-gap interface. The surface offset value was set for the first step of the power ramp in the contact definition with the values from the STAV7 output. The offset parameter is over 3 times smaller for the zero burnup as the only reasons for it are thermal expansion and relocation.
3.4 COMPARISON OF STAV7 AND ANSYS MODELS
The summary of the most important similarities and differences between the two tools is presented in Table 2. The differences in assumptions and choices made are the main cause for the discrepancies between the outcomes and they are considered the basis of the further discussion of the results.
19
Table 2: Comparison of the key assumptions and selections in STAV7 and ANSYS. Version 2 is chosen to be used in ANSYS simulation although both versions should give relatively similar results.
Calculation tool STAV7 ANSYS
Rod design An axial segment of a fuel rod A radial slice of a fuel rod segment
Pellet geometry Right circular cylinders Chamfer considered
Pellet material Perfectly stiff body Young Modulus 1011
Pellet-pellet contact Bonded Bonded
Pellet-clad contact Frictional Frictionless
Gap gas pressure Calculated by STAV7 Output from STAV7 used in ANSYS
Coolant temperature Tsat at coolant pressure
Version 1: includes the ΔT due to crud and oxide layer on the clad surface Version 2: Tsat at coolant pressure
Pellet and clad radial temperature distribution
Calculated by STAV7
Version 1: calculated by ANSYS based on the input power values Version 2: output from STAV7 used as an input to the Ansys thermal solution
Cladding thermal expansion Included in the model Neglected
Crud and oxide Temperature step jumps included in the model
Version1: manually calculated change of 𝑇1223456 Version 2: included in the temperatures imported from STAV7
Creep and swelling Included in the model Change of the pellet-clad contact surface offset according to the STAV7 output
20
4 RESULTS AND DISCUSSION
4.1 THERMAL SOLUTION
Radial temperature distribution in pellet and cladding, for different power levels, obtained from STAV7 is presented in Figure 14. Temperature curves at the two burnups have the same behaviour. The difference is that, with no irradiation effects, the temperatures at higher powers will not reach as elevated values as for the fuel with burnup. The centreline temperature for the “zero burnup” does not reach 2000 ˚C while the melting temperature is higher than for the fuel “middle life” case. It provides higher margin to CFM at a certain local power.
Figure 14: Radial temperature distribution from the pellet centerline to the cladding outer surface for different power levels, for burnup 35.6 MWd/kgU. Pellet radius is divided in equal-length sectors.
Cladding temperatures are slightly higher than temperatures for the same model found in literature. It is due to crud and oxide layer growth at the cladding outer surface which is considered in STAV7 calculations which makes it an adequate reality approximation.
4.2 STRUCTURAL SOLUTION
The static structural results from the ANSYS calculation are available for the entire 3D geometry. Strains and stresses presented in the plots, are values extracted from points close to the center of the cladding, both from the radial and axial perspective. Solutions close to the boundary condition planes are influenced by the assumptions/boundary conditions and are therefore less accurate. ANSYS results across the pellet chamfers would not be comparable with those from STAV7 as they do not include chamfering in the pellet geometry design.
The static structural results are presented for the version with liner cladding for both ANSYS and STAV7 and for the version with the cladding made entirely of the base material
21
– just for ANSYS. Only circumferential components of plastic and total strains, as well as stresses are used in the comparison. Additionally, the effective values are evaluated (according to von Mises). All the results are presented as a function of the nodal power and discussed, based on the information given in the Methodology part.
4.2.1 ANSYS vs STAV7
The comparison of results from ANSYS and STAV7 is conducted for 0.0 and 35.6 MWd/kgU burnup and the results are presented below.
4.2.1.1 Burnup 0.0 MWd/kgU
No plastic strain occurs for the fresh fuel and unirradiated cladding, neither in STAV7 nor in ANSYS simulation. Total strains (Figure 15) and stresses (Figure 16) are nearly constant for both models during the entire ramp and cladding hoop stress is negative. It implies that the pellet-clad gap remained opened at all moments of the simulations. This fact is confirmed by the STAV7 output. Effective strain values are higher than the circumferential ones – it shows the influence of axial and radial components, with the difference between them similar in STAV7 and ANSYS case. It means that even though the initial strain is different due to model differences, the behavior of the strain evolution is the same and the level of influence of the components on the remaining directions is comparable for the fresh fuel. As the plastic strain is zero and the clad radial creep is absent at zero burnup, the total strain values of around represent the elastic component and, in case of STAV7, the clad thermal expansion. This is the reason for positive value of STAV7 prediction of total circumferential strain. Clad thermal expansion is not captured in ANSYS and therefore the external pressure load on the cladding causes negative strains and stresses in circumferential direction. It is confirmed by tracking the effects of thermal expansion of Zircaloy-2 found in literature, [12], and comparing it with the difference in circumferential strains.
22
Figure 15: Total strain results for Ansys and STAV7 for the version with the liner cladding at 0.0 MWd/kgU burnup.
The stress evolution for both simulations is almost perfectly aligned during the entire ramp, with the circumferential component constant below zero and the effective stress above.
Figure 16: Stress results for ANSYS and STAV7 for the version with the liner cladding at 0.0 MWd/kgU burnup.
4.2.1.2 Burnup 35.6 MWd/kgU
At 35.6 MWd/kg U burnup, the pellet-clad gap is closed from the beginning of the simulation. Plastic strain components, presented in Figure 17, for STAV7 and ANSYS have different initial values but a similar curve evolution with power – primary constant value starts to increase at ̴ 30 kW/m. An unusual behavior is noticed for the STAV7 effective
23
plastic strain – after 40 kW/m the strain suddenly decreases, just to start increasing again and continue to follow the previous tendency.
Figure 17: Plastic strain results for ANSYS and STAV7 for the version with the liner cladding at 35.6 MWd/kgU burnup.
The behavior of total strain, presented in Figure 18, is similar to the plastic components – even though ANSYS and STAV7 values at certain powers are not the same, the evolution of the curves comparable. The major difference between the curves is a different slope and it can be a consequence of different pellet stiffness definition.
The fuel pellet has Young Modulus value defined as 1011 Pa – it is not considered to be a perfectly stiff body. It means that during expansion and interaction with the clad the pellet is being slightly compressed. It can explain the lower effective strain values in the ANSYS solution in comparison to STAV7 analysis where the pellet is treated as a perfectly rigid body.
The strain values are at first constant and start increasing before the power reaches 40 kW/m. Circumferential and effective strain curves for ANSYS are nearly perfectly aligned. It implies no effects from axial and radial directions in this simulation whereas this effect is visible in the STAV7 results. Even though the pressure load will result in axial cladding stresses, there will be much smaller in comparison with the stresses due to pellet expansion. In addition to that, the frictionless contact definition between the pellet and the clad in ANSYS does not guarantee accurate results in any direction except for the circumferential.
24
Figure 18: Total strain results for ANSYS and STAV7 for the version with the liner cladding at 35.6 MWd/kgU burnup.
The change of stress during the ramp is shown in Figure 19. The same initial values in ANSYS and in STAV7 result from the pressure and contact surface offset definitions, described in 3.3.3.2, that guarantee the same stress level at the initial power. The large difference at 60 kW/m can be a result of different constitutive models.
Figure 19: Stress results for ANSYS and STAV7 for the version with the liner cladding at 35.6 MWd/kgU burnup.
4.2.2 Liner vs no liner
The results presented in this section are obtained from the ANSYS simulation in version with and without the liner cladding at 35.6 MWd/kgU burnup. The strain and stresses obtained in the same way as in the previous comparison – as the average value from the
25
cladding – will not give sufficient results for capturing the influence of the liner. Figure 20 shows that the stress curves, for the version with and without liner, are perfectly aligned.
Figure 20: Comparison of stress changes with power for burnup 35.6 MWd/kgU.
Figure 21, Figure 22, Figure 23 and Figure 24 show the circumferential stresses in the cladding, at 60 kW/m, obtained from the ANSYS simulation. The stresses for the liner cladding are at the same level as non-liner cladding at lower powers. When the power increases, however, the difference is well observed, especially at the cladding inner surface. This difference does not occur at the outer surface, where the stress level is similar in both cases. Circumferential stresses, if high enough, can be responsible for axial/radial cladding cracks. Figure 25 presents the cladding across the pellet-pellet interface. The pellet chamfering changes the stress distribution in the cladding and creates a lower local hoop stress due to chamfer edges. As the liner material is softer than cladding base material, significantly lower stresses are observed in liner cladding.
26
Figure 21: Circumferential stress evolution in ANSYS model with the liner cladding at 60 kW/m, for 35.6 MWd/kgU burnup. Three values are extracted – at the cladding inner surface, outer surface and in between.
Figure 22: Circumferential stress evolution in ANSYS model without the liner cladding at 60 kW/m, for 35.6 MWd/kgU burnup. Three values are extracted – at the cladding inner surface, outer surface and in between.
27
Figure 23: Circumferential stress evolution in ANSYS model with the liner cladding at 60 kW/m, for 35.6 MWd/kgU burnup. The picture shows the stress values at the liner surface - in the middle of the pellet and in the chamfer area.
Figure 24: Circumferential stress evolution in ANSYS model for the cladding without the liner at 60 kW/m, for 35.6 MWd/kgU burnup. The picture shows the stress values at the cladding inner surface - in the middle of the pellet and in the chamfer area.
28
Figure 25: Circumferential stress evolution in ANSYS model with and without liner cladding at 60 kW/m, for 35.6 MWd/kgU burnup. The presented cladding radial cross section is a part across the pellet-pellet interface. The selected local stress values – at the chamfer edge and at some distance from it - show the effects of the pellet chamfer on the cladding.
5 CONCLUSIONS AND IDEAS FOR FUTURE RESEARCH
The frictionless contact between the cladding and the pellets, used as a simplification in the ANSYS model, may result in inaccurate results of stresses and strains in the axial direction. It does not, however, affect the results in circumferential direction. The discrepancies between the ANSYS and STAV7 results are the consequence of different models and assumptions of this contact area. In order to obtain more realistic results, it is advisable to create frictional contact simulations and compare the results with the frictionless assumption. A bonded contact definition between pellet and cladding is the extreme case of friction influence and is worth investigation. Introduction of the bonding contact would be a representation of a bonding layer that can be observed at high burnups and restricts the pellet movement in circumferential direction. According to literature, [3], it results in stronger stress gradients, higher local peak stresses and can affect the integrity of the pellet
29
– radial pellet cracks can occur. In addition to that, the type of connection between two pellets should be studied (use of frictional instead of bonded contact). Axial positions where two pellets touch each other are important for two reasons. Firstly, the local clad stresses close to the pellet chamfer areas are significantly greater in comparison with the mid-pellet region. Secondly, the pellet-pellet interface allows easier transport of fission products and therefore increases their amount at the clad inner surface, [4]. This phenomenon, although an essential part of SCC, cannot be directly included in a FE simulation. In order to study realistic PCI/SCC conditions, the mechanical analysis should be combined with chemical effects. The similarities in strain evolution at the higher burnup in ANSYS and in STAV7 and, at the same time, differences in the stress curve can be a consequence of the way of evaluating these physical quantities. Strain values, obtained directly, serve as the basis for stress calculation. Obtaining stresses depends on the implemented calculation models and their accuracy. Additionally, whereas the cladding material properties can be considered accurate enough, the fuel pellet material definition (as a rigid body in STAV7) could be improved to better represent the reality. Another idea for future development of this PCI project would be a more accurate pellet-clad gap properties calculation. Even when the gap is entirely closed, the gas remains in the clearance created due to pellet chamfer and in the surface pores of the materials. Changes in heat transfer due to the gas will cause steep local thermal gradient in the cladding area across the pellet-pellet interface. Therefore, a detailed gas mix composition, its pressure and thermal conductance could be included in ANSYS simulation and improve the accuracy of the results. A challenge for future researchers of the topic could be a development of a creep model in ANSYS that is built on the same principles as in STAV7. This could allow to investigate a relaxation process after a fast ramp or to simulate a different kind of ramp that involves time related material responses. Since this study covered only a simplified version of the “classical” PCI problem, the field for future research is quite wide – from the above mentioned improvements in the current model (materials, boundary conditions, time dependant simulation) to special cases of PCI, like radial pellet cracks or MPS. Moreover, a validation of the ANSYS model for PCI analysis could be performed with the use of a fuel rod ramp database from test programs.
30
6 REFERENCES
[1] Westinghouse Electric Sweden, Report BTE 14-0360, rev 0 (2013) Stress based BWR PCI Models (Westinghouse proprietary) G. Zhou
[2] Pellet Cladding Interaction Fuel Failures during Anticipated Operational Occurrences in Boiling Water Reactors (2009) J. Armijo och Z. Abdullahi
[3] EPRI (2008) Fuel Reliability Guidelines: Pellet-Cladding Interaction. Final Report. (EPRI
proprietary) [4] Kraftwerk Union AG, Nuclear Engineering and Design 65 (1981) Experimental Strategy of Fuel Perfomance Testing with respect to PCI W. Vogl, R. von Jan, H. Stehle [5] Journal of Nuclear Science and Technology (2006) Study on Incipient Cracks at Inner Surface of Cladding Liner after High Power
Irradiation Test K. Kitano, C. Losin , J. Arborelius och M. Limbäck
[6] Eight Edition (2011) Materials Science and Engineering W. Callister, D. Rethwisch [7] Journal of Nuclear Materials 172 (1990) Pellet-Clad Interaction (PCI) Failures of Zirconium Alloy Fuel Cladding – A Review B. Cox [8] Proc. Int. Semin. on Pellet-Clad Interaction in Water reactor fuels (2004) Cladding liner surface effects and PCI G. Lysell, K. Kitano, J.-E. Lindbäck and D. Schrire [9] Springer Open Journal (2014) Reduced yield stress for zirconium exposed to iodine: reactive force field simulation M. Rossi, C. Taylor, A. van Duin [10] Reactor Physics Division, KTH (2015) Chemistry and Physics of Nuclear Fuels (course) Lecture 10 – The composition of spent fuel M. Jolkkonen [11] http://www.ansys.stuba.sk/html/guide_55/g-the/GTHE2.htm, last accessed
28/05/2016 [12] Westinghouse Electric Sweden, Report BU 97-091, rev 3 (2007) STAV7 Model description (Westinghouse proprietary) G.Zhou
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Appendix A FUEL PELLET MATERIAL PROPERTIES Fractional linear thermal expansion of UO2, below its melting point, in function of temperature, ∆8
89𝑇 , is obtained according to Westinghouse norms.
For ∆889
<< 1 and for small changes of the linear thermal expansion coefficient (𝛼6; [℃>?]) with
temperature, it can be estimated as:
𝛼6; = ∆𝐿/𝐿D𝛥𝑇
Figure 26 shows the thermal expansion coefficient in function of temperature.
Figure 26: Pellet thermal expansion coefficient changes in temperature.
Thermal conductivity of the fuel, in function of temperature and local burnup, is calculated based on the porosity coefficient and the fraction of theoretical density. For the burnup of 40 MWd/kgU, the thermal conductivity becomes as presented in Figure 27.
0,0E+00
4,0E-‐06
8,0E-‐06
1,2E-‐05
1,6E-‐05
Ther
mal
exp
ansi
on
coef
fici
ent [
1/°C
]
Temperature [°C]0.0E+00
4.0E-06
8.0E-06
1.2E-05
1.6E-05
Ther
mal
exp
ansi
on c
oeff
icie
nt [1
/°C
]
Temperature [°C]
32
Figure 27: Thermal conductivity as a function of temperature for fresh fuel and 35.6 MWd/kgU burnup.
0
5
10
15
20
25
0 500 1000 1500 2000 2500
Ther
mal
con
duct
ivit
y [W
/m/K
]
Temperature [°C]
0.0 MWd/kgU
35.6 MWd/kgU
33
Appendix B CLADDING MATERIAL PROPERTIES Base material properties are estimated using cladding thermal mechanical model. The flux is considered as 1023 5
FG while the cold work and oxidation effects are neglected.
The strain-stress relation in the elastic region can be described as:
𝜎 = 𝜀 ∙ 𝐸
and in the plastic region as:
𝜎 = 𝐾 ∙ 𝜀5 ∙ (1000𝜀)F
where: 𝜎 − stress [Pa], 𝐸 – Young Modulus [Pa], 𝜀 − total strain, 𝜀 − strain rate [s-1], 𝜀 = 1.5 ∙ 10>* s-1, 𝐾 − strength coefficient, 𝑛 − strain hardening exponent, 𝑚 − strain rate exponent. The coefficient K and exponents n and m are calculated in accordance with the Westinghouse models. The strength coefficient is obtained in function of temperature, using a relation valid for 𝑇 ≤ 730 𝐾. The strain hardening exponent is first evaluated for an unirradiated cladding and 𝑇 ≤ 850 𝐾. Then the exponent changes due to irradiation are taken into consideration and a new coefficient is estimated by employing a relation for fluences Φ ≤ 2 ∙ 10&* 5
FG . The value of the strain rate exponent is chosen for 𝑇 ≤ 730 𝐾 and does not change with irradiation. The yield strength, Y, is calculated using the following relations:
𝑌 = 𝑎?/(?>5)
𝑎 = 𝐾 ∙ (1000𝜀)F ∙ 𝐸>?
The strain-stress relation for irradiated zircaloy in different temperatures, as well as the yield strength values, are presented in Figure 28 andFigure 29.
34
Figure 28: Strain – stress relation for irradiated Zircaloy-2 in different temperatures.
In order to estimate strain-stress correlation for the liner material after irradiation, the change of its hardness is considered. As mentioned in paragraph 2.2, hardness is directly proportional to the yield strength of the material. The increase in hardness was measured in Barsebäck 2 reactor and described in a report by Studsvik Nuclear AB. Yield strength values evaluated for different temperatures are presented in Figure 29.
Figure 29: Base material and liner yield strength changes with temperature.
0
200
400
600
800
Stre
ss [M
Pa]
Strain
20˚C
100˚C
200˚C
300˚C
400˚C
180
280
380
480
580
680
Yie
ld s
tren
gth
[MP
a]
Temperature [˚C]
Base material
Liner
35
TRITA FYS 2016:38 ISSN 0280-316X ISRN KTH/FYS/--16:38—SE